Is it acceptable to ask an unknown professor for help in a related field during his office hours?

You can certainly ask a professor politely if he/she would be willing to talk to you about some research topic that might be of mutual interest to you. Don't be too surprised if the professor isn't interested in discussing it with you though- what's interesting to you might well not be interesting to that professor.

Furthermore, it is unreasonable to expect a professor to help you with a class that you are taking from some other professor or in a different academic department. You have an instructor for that other course, and you probably also have access to a tutoring center where you can get help from people whose job it is to provide tutoring.

Ordinarily, I'd say go ahead and ask, but both the examples you cite sound like you should not need to ask an outside professor. The first is clearly the responsibility of the prof of your engineering class -- AFTER you've done all you can do to learn the math on your own -- and the second seems like enough time in front of a search engine should do the job.

To expand a little bit, for the question about the apache log, it doesn't sound like you're anywhere near prepared enough to even ask the question. Spend an hour trying to find your own answer. Do some prelim research to better inform your question. If you haven't found the answer, you might consider looking for the appropriate Stack Exchange group, and asking there. If you still haven't gotten the answer you need, now would be the time to try to find an expert to talk to.

Although I certainly do aim to be civil to students who "drop in", and, yes, am able to gauge earnestness... : the point is that anything that comes up in undergrad engineering or undergrad math classes or first or second-year grad courses is (almost surely) standard. That is, many sources exist for it, on-line and off. That does not entail that an experienced expert can't add any insights to the standard sources, but it does mean that a student who hasn't looked at (or found...) the standard sources is asking (probably inadvertently) for tooooo much help. That is, very likely looking at any one of the standard sources would instantly resolve the question, and truly expert insights are irrelevant and unnecessary (and a waste of an expert's time).

One advantage of on-line sites like MathStackExchange is the asynchronousness: I only look at them when I'm in the mood, and no one is offended if I don't respond at all, and, indeed, I can give whatever response I want and then I'm done.

In that regard, I almost prefer drop-ins to appointments, since, in fact, appointments consume more mental resources than spur-of-the-moment things. Nevertheless, of course, all my time is planned out for nearly every day. Thus, "drop in" questions by email are preferable to in-person. But/and, yes, my email response may be that this is standard and one can easily look it up... e.g., in notes I've written that are on-line.

The worst-case scenario is non-math people (e.g., engineering students, sorry) vastly under-estimating the effort often required to really prove things, if that's what they believe that they want. It is a happy miracle that mathematics works so well, especially at "modest scales" in the physical world, and, mostly, it is easy to obtain physical corroboration for heuristic mathematics. That is, plausibility arguments are often straightforward, even if requiring some wishful thinking. Part of the "miracle" is that, historically, it has often proven difficult to distill a physical narrative into genuine mathematical terms that no longer depend on literal physical intuition for their sense or proof. (Sure, for physical scientists, there may indeed be no reason to even attempt to abstract mathematics from the physical.)

If you send email to a professor, introduce yourself properly and politely, compose it nicely, and don't be offhanded about it. Otherwise the professor might not take you seriously and ignore the email. I had an otherwise very smart student doing research on land use who wanted to meet professors working in the field. She'd write something like, "hey, could I come meet you?" and then wonder why she got no reply. I had to teach her in etiquette as well as the subject.